Measurements of jet substructure are key to probing the energy frontier at colliders, and many of them use track-based observables which take advantage of the angular precision of tracking detectors. Theoretical calculations of track-based observables require `track functions', which characterize the transverse momentum fraction $r_q$ carried by charged hadrons from a fragmenting quark or gluon. This letter presents a direct measurement of $r_q$ distributions in dijet events from the 140 fb$^{-1}$ of proton--proton collisions at $\sqrt{s}=13$ TeV recorded with the ATLAS detector. The data are corrected for detector effects using machine-learning methods. The scale evolution of the moments of the $r_q$ distribution is sensitive to non-linear renormalization group evolution equations of QCD, and is compared with analytic predictions. When incorporated into future theoretical calculations, these results will enable a precision program of theory-data comparison for track-based jet substructure observables.
$r_{q}$, Gluon jets, $240\text{GeV} \leq p_T < 300~\text{GeV}$, Gluon $\eta$, Fig 5
$r_{q}$, Gluon jets, $300~\text{GeV} \leq p_T < 400~\text{GeV}$, Gluon $\eta$, Fig 5
$r_{q}$, Gluon jets, $400~\text{GeV} \leq p_T < 500~\text{GeV}$, Gluon $\eta$, Fig 5
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